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Spatial multi-level interacting particle simulations and information theory-based error quantification
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Remarks on the Contributions of Constantine M. Dafermos to the Subject of Conservation Laws
Get PDFThis is an expository article highlighting certain of the contributions of Constantine M. Dafermos on the subject of Conservation Law
Reflection and refraction of an Airy beam at a dielectric interface
Get PDFReflection and refraction of a finite-power Airy beam at the interface between two dielectric media are investigated analytically and numerically. The formulation takes into account the paraxial nature of the optical beams to derive convenient field evolution equations in coordinate frames moving along Snell’s refraction and reflection axes. Through numerical simulations, the self-accelerating dynamics of the Airy-like refracted and reflected beams are observed. Of special interest are the cases of critical incidence at Brewster and total-internal-reflection (TIR) angles. In the former case, we find that the reflected beam achieves to self-heal despite the severe supression of a part of its spectrum, while, in the latter case, the beam remains nearly unaffected except for the Goos-Hänchen shift. The self-accelerating quality persists even if the beam is trapped by multiple TIRs inside a dielectric film. Grazing incidence of an Airy beam at the interface between two media with close refractive indices is also investigated revealing that the interface can act as a filter cepending on the beam scale and tilt. We finally consider reverse refraction and perfect imaging of an Airy beam into a left-handed medium
Macroscopic limits and phase transition in a system of self-propelled particles
Get PDFWe investigate systems of self-propelled particles with alignment interaction. Compared to previous work [13, 19], the force acting on the particles is not normalized and this modification gives rise to phase transitions from disordered states at low density to aligned states at high densities. This model is the space inhomogeneous extension of [20] in which the existence and stability of the equilibrium states were investigated. When the density is lower than a threshold value, the dynamics is described by a non-linear diffusion equation. By contrast, when the density is larger than this threshold value, the dynamics is described by a similar hydrodynamic model for self-alignment interactions as derived in [13, 19]. However, the modified normalization of the force gives rise to different convection speeds and the resulting model may lose its hyperbolicity in some regions of the state space
The ground state of a Gross–Pitaevskii energy with general potential in the Thomas–Fermi limit
Get PDFWe study the ground state which minimizes a Gross–Pitaevskii
energy with general non-radial trapping potential, under the unit mass constraint, in the Thomas–Fermi limit where a small parameter tends to 0. This ground state plays an important role in the mathematical treatment of recent
experiments on the phenomenon of Bose–Einstein condensation, and in the study of various types of solutions of nonhomogeneous defocusing nonlinear Schrodinger equations. Many of these applications require delicate estimates
for the behavior of the ground state near the boundary of the condensate, as the singular parameter tends to zero, in the vicinity of which the ground state has irregular behavior in the form of a steep corner layer. In particular, the role of this layer is important in order to detect the presence of vortices in the small density region of the
condensate, understand the superfluid flow around an obstacle, and also has a leading order contribution in the energy. In contrast to previous approaches, we utilize a perturbation argument to go beyond the classical Thomas–Fermi
approximation and accurately approximate the layer by the Hastings–McLeod solution of the Painleve–II equation. This settles an open problem, answered very recently only for the special case of the model harmonic potential. In fact, we even improve upon previous results that relied heavily on the radial symmetry of the potential trap. Moreover, we show that the ground state has the maximal regularity available,
namely it remains uniformly bounded in the
1/2-Holder norm, which is the exact Holder regularity of the singular limit profile, as the singular parameter tends to zero. Our study is highly motivated by an interesting open problem posed recently by Aftalion, Jerrard, and Royo-Letelier, and an open question of Gallo and Pelinovsky,
concerning the removal of the radial symmetry assumption from the potential trap
Band-specific phase engineering for curving and focusing light in waveguide arrays
Get PDFBand specific design of curved light caustics and focusing in optical waveguide arrays is introduced. Going beyond the discrete, tight-binding model, which we examined recently, we show how the exact band structure and the associated diffraction relations of a periodic waveguide lattice can be exploited to phase-engineer caustics with predetermined convex trajectories or to achieve optimum aberration-free focal spots. We numerically demonstrate the formation of convex caustics involving the excitation of Floquet-Bloch modes within the first or the second band and even multi-band caustics created by the simultaneous excitation of more than one bands. Interference of caustics in abruptly autofocusing or collision scenarios are also examined. The experimental implementation of these ideas should be straightforward since the required input conditions involve phase-only modulation of otherwise simple optical wavefronts. By direct extension to more complex periodic lattices, possibilities open up for band specific curving and focusing of light inside 2D or even 3D photonic crystals
Global-in-time behavior of the solution to a Gierer-Meinhardt system
Get PDFGierer-Meinhardt system is a mathematical model to describe biological pattern formation due to activator and inhibitor. Turing pattern is
expected in the presence of local self-enhancement and long-range inhibition.
The long-time behavior of the solution, however, has not yet been clarified mathematically. In this paper, we study the case when its ODE part takes
periodic-in-time solutions, that is, . Under some additional assumptions on parameters, we show that the solution exists global-in-time and absorbed into one of these ODE orbits. Thus spatial patterns eventually dis- appear if those parameters are in a region without local self-enhancement or long-range inhibition
Resonance Properties of Optical All-Dielectric Metamaterials Using Two-Dimensional Multipole Expansion
Get PDFWe examine the electromagnetic response of metamaterial unit elements consisting of dielectric rods embedded in a non-magnetic background medium. We establish a theoretical framework where the response is described through the electric and magnetic multipole moments that are simultaneously generated via the polarization currents that are excited upon the incidence of plane waves. The corresponding dipole and quadrupole polarizabilities are then calculated as a function of the Mie scattering coefficients, and their resonances are mapped for the case of dielectric cylindrical rods as a function of the geometry and the material parameters utilized. The results provide critical insight on the anisotropic response of two-dimensional rod-type metamaterials and can be used as a unified methodology in the calculation of exotic effective electromagnetic parameters involved in phenomena such as optical magnetism